Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders
CCOURANT-SHARP EIGENVALUES OF COMPACT FLATSURFACES: KLEIN BOTTLES AND CYLINDERS
PIERRE BÉRARD, BERNARD HELFFER, AND ROLA KIWAN
Abstract.
The question of determining for which eigenvalues there exists aneigenfunction which has the same number of nodal domains as the label of theassociated eigenvalue (Courant-sharp property) was motivated by the analysis ofminimal spectral partitions. In previous works, many examples have been ana-lyzed corresponding to squares, rectangles, disks, triangles, tori, Möbius strips,. . . .A natural toy model for further investigations is the flat Klein bottle, a non-orientable surface with Euler characteristic 0, and particularly the Klein bottleassociated with the square torus, whose eigenvalues have higher multiplicities. Inthis note, we prove that the only Courant-sharp eigenvalues of the flat Klein bottleassociated with the square torus (resp. with square fundamental domain) are thefirst and second eigenvalues. We also consider the flat cylinders (0 , π ) × S r where r ∈ { . , } is the radius of the circle S r , and we show that the only Courant-sharpDirichlet eigenvalues of these cylinders are the first and second eigenvalues. Introduction
Given a compact Riemannian surface (
M, g ), we write the eigenvalues of the Laplace-Beltrami operator − ∆ g ,(1.1) λ ( M ) < λ ( M ) ≤ λ ( M ) ≤ . . . , in nondecreasing order, starting from the label 1, with multiplicities accounted for.If the boundary ∂M of M is non-empty, we consider Dirichlet eigenvalues.Courant’s nodal domain theorem (1923) states that any eigenfunction associatedwith the eigenvalue λ k has at most k nodal domains (connected components of thecomplement of the zero set of u ). The eigenvalue λ k is called Courant-sharp ifthere exists an associated eigenfunction with precisely k nodal domains. It followsfrom Courant’s theorem that the eigenvalues λ and λ are Courant-sharp, andthat λ k − < λ k whenever λ k is Courant-sharp. Courant-sharp eigenvalues appearnaturally in the context of partitions, [13].Back in 1956, Pleijel proved that there are only finitely many Courant-sharp eigen-values. More precisely, Pleijel’s original proof [24] applied to Dirichlet eigenvaluesof bounded domains in R . It was later adapted to more general domains ([23]) andto general closed Riemannian manifolds ([6]).Courant-sharp eigenvalues of flat tori and Möbius strips were studied in [12, 19, 3]and [5] respectively. The purpose of this note is to look at the other compact flatsurfaces, Klein bottles and cylinders, and to prove the following theorems. Date : September 15, 2020 (BHK-klein-cylinder-200914.tex).2010
Mathematics Subject Classification.
Key words and phrases.
Spectral theory, Courant theorem, Laplacian, Nodal sets, Klein bottle,Cylinder. a r X i v : . [ m a t h . SP ] S e p P. BÉRARD, B. HELFFER, AND R.KIWAN
Theorem 1.1.
Let K denote the flat Klein bottle associated with the square torus,resp. K the flat Klein bottle whose fundamental domain is a square. Then the onlyCourant-sharp eigenvalues of K c , c ∈ { , } , are λ and λ . Theorem 1.2.
Let C r denote the flat cylinder (0 , π ) × S r , with the product metric.Here S r denotes the circle with radius r . Then, for r ∈ n , o , the only Courant-sharp Dirichlet eigenvalues of C r , are λ and λ . Figure 1.1.
The Klein bottles K c (left) and the cylinders C r (right).Courant-sharp eigenvalues have previously been determined for several compact sur-faces. We refer to the following papers and their bibliographies. (cid:5) Closed surfaces: round 2-sphere and projective plane, [22]; flat tori, [12, 19,3]. (cid:5)
Plane domains and compact surfaces with boundary (different boundary con-ditions might be considered): square, [24, 2, 15, 9, 10]; equilateral triangle,[3]; 2-rep-tiles, [1]; thin cylinders, [11]; Möbius strips, [5].There are also a few results in higher dimensions, see for example [16, 20, 14].Most of the papers mentioned above adapt the method introduced by Pleijel in [24]to the example at hand.The note is organized as follows. In Section 2, we recall the main lines of Pleijel’smethod. More precisely, we show how a lower bound on the ratio λ k ( M ) k and a lowerbound on the Weyl counting function can be used to restrict the search for Courant-sharp eigenvalues to a finite set of eigenvalues. To actually determine the Courant-sharp eigenvalues one then needs to analyze the nodal patterns of eigenfunctions ina finite set of eigenspaces. In Section 3, we recall some basic facts concerning Kleinbottles. In Sections 4 and 5, we adapt Pleijel’s method (Section 2) to the flat Kleinbottles K and K . In Section 6, we adapt Pleijel’s method (Section 2) to the flatcylinders C r , with r ∈ n , o .2. Pleijel’s method summarized
In order to prove that the number of Courant-sharp (Dirichlet) eigenvalues of acompact Riemannian surface (
M, g ) is finite, one only needs two ingredients,(1) the classical Weyl asymptotic law, W M ( λ ) = | M | π λ + O ( √ λ ), for the countingfunction W M ( λ ) = { j | λ j ( M ) < λ } , where | M | denotes the area of M ; OURANT-SHARP EIGENVALUES 3 (2) an asymptotic inequality à la Faber-Krahn for domains ω ⊂ M with smallenough area,(2.1) δ ( ω ) ≥ (1 − ε ) πj , | ω | provided that | ω | ≤ C ( M, ε ) | M | , for some ε ∈ [0 , C ( M, ε ). Here, δ ( ω ) denotes theleast Dirichlet eigenvalue of ω , and j , is the first positive zero of the Besselfunction J ( j , ≈ . − ε ) δ ( ω ∗ ), where ω ∗ denotesa disk in R with area equal to | ω | . The existence of such an inequality (for agiven (cid:15) ∈ (0 , W M ( λ ) ≥ | M | π λ − A ( M ) √ λ − B ( M ) , for all λ ≥ A ( M ) and B ( M ) depending on the geometry of ( M, g ). Lemma 2.1.
Assume that ( M, g ) satisfies (2.1) , with ε < − j , , and (2.2) . Let λ k ( M ) be a Courant-sharp eigenvalue with k ≥ C ( M,ε ) . Then, λ k ( M ) k ≥ (1 − ε ) πj , | M | , (2.3) | M | (1 − ε ) πj , λ k ( M ) ≥ k ≥ | M | π λ k ( M ) − A ( M ) q λ k ( M ) − B ( M ) + 1 , (2.4) In particular, F M,ε ( λ k ( M )) ≤ , where (2.5) F M,ε ( λ ) = | M | π − − ε ) j , ! λ − A ( M ) √ λ − B ( M ) + 1 , so that q λ k ( M ) ≤ D ( M, ε ) , where D ( M, ε ) is the largest root of the equation F M,ε ( λ ) = 0 .Proof. To prove (2.3), choose any eigenfunction associated with λ k ( M ) and havingprecisely k nodal domains. Since k ≥ C ( M,ε ) , one of them, call it ω , has area | ω | ≤ | M | k ≤ C ( M, ε ) | M | , so that its first Dirichlet eigenvalue satisfies (2.1). Then,use the fact that λ k ( M ) = δ ( ω ).To prove (2.4), use the fact that k − W M ( λ k ( M )) (this is because the eigenvalueis Courant-sharp), and apply inequalities (2.2) and (2.3).Since j , >
2, choosing ε < − j , , the coefficient of the leading term in F M,ε ispositive, and the function tends to infinity when λ tends to infinity. (cid:3) P. BÉRARD, B. HELFFER, AND R.KIWAN
Corollary 2.2.
For ε < − j , , in order to be Courant-sharp, the eigenvalue λ k ( M ) must satisfy λ k − ( M ) < λ k ( M ) , and (2.6) either k < C ( M,ε ) , or k ≥ C ( M,ε ) , q λ k ( M ) ≤ D ( M, ε ) , and λ k ( M ) k ≥ (1 − ε ) πj , | M | . To conclude whether the eigenvalue λ k ( M ) is actually Courant-sharp, it remainsto determine the maximum number of nodal domains of an eigenfunction in theeigenspace E ( λ k ( M )) . Remarks 2.3. (1) In Sections 4 and 6, we will use the fact that we can choose ε = 0 in theisoperimetric inequality (2.1) for the flat Klein bottles and for the flat cylin-ders. This follows from [17] , and was already used in [19, 3, 5] for the flattorus and for the Möbius band.(2) When ∂M = ∅ , and for Neumann or Robin eigenfunctions of ( M, g ) , theinequality (2.1) can only be applied to a nodal domain which does not touchthe boundary ∂M . To prove a result à la Pleijel for Neumann or Robineigenfunctions, as in [25, 21, 15, 9, 10] , it is necessary to take care of thisdifficulty. Preliminaries on Klein bottles
Klein bottles.
In this note, we are interested in the flat Klein bottles . Moreprecisely, given a, b >
0, we consider the isometries of R given by(3.1) τ : ( x, y ) ( x, y + b ) ,τ : ( x, y ) ( x + a, y ) ,τ : ( x, y ) ( x + a , b − y ) . We denote by G (resp. G ) the group generated by τ and τ (resp. by τ and τ ).These groups act properly and freely by isometries on R equipped with the usualscalar product. Since τ = τ , the group G is a subgroup of index 2 of the group G . We denote by T a,b (resp. K a,b ) the torus R /G (resp. the Klein bottle R /G ).We equip T a,b and K a,b with the induced flat Riemannian metrics.A fundamental domain for the action of G (resp. G ) on R is the rectangle T a,b =(0 , a ) × (0 , b ) (resp. the rectangle K a,b = (0 , a ) × (0 , b ), see Figure 3.1 (A)). Thehorizontal sides of K a,b are identified with the same orientation, the vertical sidesare identified with the opposite orientations.The geodesics of the Klein bottle are the images of the lines in R under the Rie-mannian covering map R → K a,b . They can be looked at in the fundamental domain K a,b , taking into account the identifications ( x, ∼ ( x, b ) and (0 , y ) ∼ ( a , b − y ).Among them, we have some special geodesics, see Figure 3.1 (B)-(C), (cid:5) t ( t,
0) and t ( t, b ), for 0 ≤ t ≤ a , which are periodic geodesics oflength a ; For the classification of complete, flat surfaces, we refer to [18, p. 222-223], or [26, Chap. 2.5].A summary is given in Appendix A.
OURANT-SHARP EIGENVALUES 5 (cid:5) for 0 < y < b , γ y : t ( ( t, y ) , ≤ t ≤ a , ( t − a , b − y ) , a ≤ t ≤ a, which is a periodic geodesic of length a ; the two horizontal lines in blue inFigure 3.1 (C) yield a periodic geodesic of the Klein bottle; (cid:5) for 0 ≤ x ≤ a , t ( x , t ), with 0 ≤ t ≤ b , is a periodic geodesic of length b . Remark 3.1.
The description of geodesics of the Klein bottles as projected linesimplies that the shortest, nontrivial, periodic geodesic of K a,b has length min n a , b o . Remark 3.2.
Scissoring the Klein bottle along the blue geodesic t γ y ( t ) , with < y < b and ≤ t ≤ a , divides the surface into two Möbius strips whose soulsare the geodesics t ( t, and t ( t, b ) , see Figure 3.1 (D). The isometry τ of R induces an isometry on the torus T a,b so that we can identify K a,b with the quotient T a,b / { Id, τ } . It follows that the eigenfunctions of the Kleinbottle K a,b are precisely the eigenfunctions of the torus T a,b which are invariant underthe map τ . Because τ is orientation reversing, the surface K a,b is non-orientable withorientation double cover T a,b . (a) (b)(c) (d) Figure 3.1.
Fundamental domain, geodesics, partition into Möbius strips3.2.
The spectrum of Klein bottles.
A complete family of (complex) eigenfunc-tions of the flat torus T a,b is(3.2) f m,n ( x, y ) = exp (cid:18) i πmxa (cid:19) exp (cid:18) i πnyb (cid:19) , m, n ∈ Z , P. BÉRARD, B. HELFFER, AND R.KIWAN with associated eigenvalues ˆ λ ( m, n ) = 4 π (cid:16) m a + n b (cid:17) . Given some eigenvalue λ of T a,b , we introduce the set,(3.3) L λ = n ( m, n ) ∈ Z | ˆ λ ( m, n ) = λ o A general (complex) eigenfunction of T a,b , with eigenvalue λ , is of the form(3.4) φ = X ( m,n ) ∈L λ α m,n f m,n , with α m,n ∈ C .The function φ in invariant under τ , φ = φ ◦ τ , if and only if X ( m,n ) ∈L λ α m,n f m,n ( x, y ) = X ( m,n ) ∈L λ α m,n ( − m f m, − n ( x, y ) , or, equivalently, if and only if(3.5) α m, − n = ( − m α m,n , ∀ ( m, n ) ∈ L λ . We can rewrite a τ -invariant eigenfunction φ as,(3.6) φ = X ( m, ∈L λ ,m even α m, f m, + X ( m,n ) ∈L λ ,n> α m,n ( f m,n + ( − m f m, − n ) . The following lemma follows readily.
Lemma 3.3.
A complete family of real eigenfunctions of the flat Klein bottle K a,b is given by the following functions. (3.7) For m = 0 , n ∈ N : cos (cid:16) πnyb (cid:17) ; for m ∈ N • even , n ∈ N : cos (cid:16) πmxa (cid:17) cos (cid:16) πnyb (cid:17) ; sin (cid:16) πmxa (cid:17) cos (cid:16) πnyb (cid:17) ; for m ∈ N • odd , n ∈ N • : cos (cid:16) πmxa (cid:17) sin (cid:16) πnyb (cid:17) ; sin (cid:16) πmxa (cid:17) sin (cid:16) πnyb (cid:17) . Here, N denotes the set of non-negative integers, and N • the set of positive integers. Remark 3.4. If L λ ∩ ( { }× Z ) = ∅ , the multiplicity of λ is odd; if L λ ∩ ( { }× Z ) = ∅ ,the multiplicity of λ is even. Choices for a and b . In this paper, we restrict our attention to the case a = b = 2 π , i.e. to the flat Klein bottle K := K π, π , whose fundamental domain isthe rectangle K = (0 , π ) × (0 , π ), and to the case a = 2 π , b = π , i.e., to the flat Kleinbottle K := K π,π , whose fundamental domain is the square K = (0 , π ) × (0 , π ).As in [12] for flat tori, we could consider other values of the pair ( a, b ), in particularwe could look at what happens when a is fixed, b tends to zero, and vice-versa.We denote the associated square flat tori by T and T respectively, with corre-sponding fundamental domains T = (0 , π ) × (0 , π ) and T = (0 , π ) × (0 , π ).As points of the spectrum, the eigenvalues of the flat Klein bottle K c , c ∈ { , } arethe numbers of the form ˆ λ ( p, q ) = p + c q , with p, q ∈ N , and the extra conditionthat p is even when q = 0. As usual, the eigenvalues of K c are listed in nondecreasingorder, multiplicities accounted for, starting from the label 1,(3.8) 0 = λ ( K c ) < λ ( K c ) ≤ λ ( K c ) ≤ · · · . For λ ≥
0, we introduce the
Weyl counting function ,(3.9) W K c ( λ ) = { j | λ j ( K c ) < λ } . OURANT-SHARP EIGENVALUES 7
Weyl’s asymptotic law tells us that(3.10) W ( λ ) = | K c | π λ + O ( √ λ ) = π c λ + O ( √ λ ) , where | K c | denotes the area of the Klein bottle, namely | K c | = π c .For later purposes, we also introduce the set(3.11) L c ( λ ) := n ( m, n ) ∈ N | m + c n < λ o and the counting function ,(3.12) L c ( λ ) = (cid:16)n ( m, n ) ∈ N | m + c n < λ o(cid:17) . Courant-sharp eigenvalues of the Klein bottles K c , c ∈ { , } The purpose of this section is to determine the Courant-sharp eigenvalues of theKlein bottles K c , following Pleijel’s method, Section 2. Lemma 4.1.
Let K c , c ∈ { , } be the flat Klein bottles introduced in Section 3. Let λ k ( K c ) be a Courant-sharp eigenvalue of K c , with label k ≥ πc (i.e. k ≥ when c = 1 ; k ≥ when c = 2 ).When c = 1 , λ k ( K ) k ≥ j , π ≥ . , (4.1) 2 πj , λ k ( K ) ≥ k ≥ π λ k ( K ) − q λ k ( K ) − . (4.2) When c = 2 , λ k ( K ) k ≥ j , π ≥ . , (4.3) πj , λ k ( K ) ≥ k ≥ π λ k ( K ) − q λ k ( K ) − . (4.4) In particular, λ k ( K ) < and λ k ( K ) ≤ .Proof. By Remark 3.1, the shortest closed geodesic of K c , c ∈ { , } , has length π . According to [17, § 7], any domain ω ⊂ K c , with area | ω | ≤ π , satisfies theEuclidean isoperimetric inequality | ∂ω | ≥ π | ω | (where | ∂ω | denotes the length ofthe boundary ∂ω ). It follows that δ ( ω ) ≥ πj , | ω | . If λ k ( K c ) is Courant-sharp with k ≥ πc , then, for any associated eigenfunction u with precisely k nodal domains, there exists at least one nodal domain ω with | ω | ≤ K c k ≤ π , and we have λ k ( K c ) = δ ( ω ) ≥ πj , | ω | ≥ k cj , π . This proves inequalities (4.1) and (4.3).To prove the other inequalities, consider the set E c ( λ ) := n ( x, y ) | ≤ x, y, x + c y < λ o . P. BÉRARD, B. HELFFER, AND R.KIWAN
Then, E c ( λ ) ⊂ [ ( m,n ) ∈L c ( λ ) [ m, m + 1] × [ n, n + 1] , and hence(4.5) L c ( λ ) = L c ( λ )) ≥ |E c ( λ ) | = π c λ . From the description of the spectrum of K c (Lemma 3.3), the contribution of a pair( m, n ) ∈ L c ( λ ), to W K c ( λ ) isa) 2 if m, n ≥ m = 0, and n ≥ m ≥ n = 0;d) 0 otherwise.Recall that multiplicities also arise from the number of solutions of the equation m + c n = m + c n . It follows that, W K c ( λ ) = 2 L c ( λ ) − b√ λ c − b √ λc c − ! + b √ λc c + 1 ! + 2 b √ λ c , = 2 L c ( λ ) − b√ λ c − b √ λc c + 2 b √ λ c − , where b x c denotes the integer part of x ≥
0. It follows that, for all λ ≥ W K c ( λ ) ≥ π λ − √ λ − c = 1 , π λ − √ λ − c = 2 . Inequalities (4.2) and (4.4) follow from the previous inequalities. (cid:3)
The following tables display the eigenvalues of K c less than or equal to 25 ( c = 1),resp. 47 ( c = 2), the corresponding labeled eigenvalues, and the ratio λ k min ( K c ) k min which should be larger than or equal to j , π ≈ . λ k min ( K ) is Courant-sharp ,j , π ≈ . λ k min ( K ) is Courant-sharp . Since the eigenvalues λ ( K c ) , λ ( K c ) are Courant-sharp, we conclude from Table 4.1that Theorem 1.1 is proved in the case c = 2. To finish the proof of the theoremwhen c = 1, it remains to investigate λ ( K ) and λ ( K ). This is done in the nextsection. OURANT-SHARP EIGENVALUES 9 K λ k min ( K ) λ k max ( K ) λ k min ( K ) k min λ λ —2 λ λ —4 λ λ —5 λ λ . λ λ . λ λ . λ λ . λ λ . λ λ . λ λ . λ λ . λ λ . λ λ . K λ k min ( K ) λ k max ( K ) λ k min ( K ) k min λ λ —5 λ λ λ λ . λ λ . λ λ . λ λ . λ λ . λ λ . λ λ . λ λ . λ λ . λ λ . λ λ . λ λ . λ λ . λ λ . Table 4.1.
The first eigenvalues of K (left) and K (right)5. The eigenvalues λ ( K ) and λ ( K ) are not Courant-sharp In order to finish the proof of Theorem 1.1 for the Klein bottle K , we investigatethe eigenvalues λ ( K ) = ˆ λ (1 ,
1) and λ ( K ) = ˆ λ (2 ,
0) = ˆ λ (0 , The eigenvalue λ ( K ) is not Courant-sharp. A general eigenfunction as-sociated with λ has the form ( A cos( x ) + B sin( x )) sin( y ). It is sufficient to look ateigenfunctions of the form sin( x − α ) sin( y ). These eigenfunctions have exactly twonodal domains in K . It follows that λ is not Courant-sharp, see Figure 5.1.5.2. The eigenvalue λ ( K ) is not Courant-sharp. A general eigenfunction as-sociated with λ has the form A cos(2 x )+ B sin(2 x )+ C cos(2 y ). Up to multiplicationby a scalar, it suffices to consider the family cos θ cos(2 x − α ) + sin θ cos(2 y ), with θ ∈ [0 , π ) and α ∈ [0 , π ). Choosing the fundamental domain appropriately, we canassume that α = 0. Changing y to y + π , we see that it suffices to consider θ ∈ [0 , π ].The nodal sets are known explicitly when θ = 0 or π . We now consider the family(5.1) φ θ ( x, y ) = cos θ cos(2 x ) + sin θ cos(2 y ) , θ ∈ (0 , π . Figure 5.1.
Nodal domains of sin( x − α ) sin( y )The critical zeros (points at which both the function and its differential vanish)satisfy the system,(5.2) cos θ cos(2 x ) + sin θ cos(2 y ) = 0 , sin(2 x ) = 0 , sin(2 y ) = 0 . It follows that critical zeros only occur for θ = π , so that the nodal set of φ θ is aregular curve when θ = π . The nodal set of the eigenfunction(5.3) φ π ( x, y ) = cos(2 x ) + cos(2 y ) = 2 cos( x + y ) cos( x − y )is explicit, see Figure 5.2 (B). An analysis à la Stern, see [5, Section 5] or [4], showsthat the nodal sets of φ θ for 0 < θ < π and π < θ < π are given by Figures 5.2 (A)and (C) respectively. More precisely, we first note that the common zeros to φ and φ π are common zeros to all φ θ . Since θ ∈ (0 , π ), and hence sin θ cos θ >
0, itfollows that, except for the common zeros, the nodal set of φ θ is contained in theset cos(2 x ) cos(2 y ) <
0. Finally, depending on the sign of π − θ , we can use thenodal set of cos(2 x ) or the nodal set of cos(2 y ) as barrier to obtain the behaviourdescribed in the figures.It follows that an eigenfunction associated with λ has at most 4 nodal domains in K , and hence that λ ( K ) is not Courant-sharp. (a) (b) (c) Figure 5.2.
Nodal sets for cos θ cos(2 x ) + sin θ cos(2 y )This completes the proof of Theorem 1.1 in the case c = 1. OURANT-SHARP EIGENVALUES 11 Courant-sharp Dirichlet eigenvalues of flat cylinders
Preliminaries.
Given r >
0, let C r = (0 , π ) × S r denote the cylinder withradius r and length π , equipped with the flat product metric. We consider theDirichlet boundary condition on ∂ C r = { , π } × S r . A complete family of Dirichleteigenfunctions is given by,(6.1) sin( mx ) cos (cid:16) nyr (cid:17) , for m ∈ N • , n ∈ N , sin( mx ) sin (cid:16) nyr (cid:17) , for m ∈ N • , n ∈ N • , with associated eigenvalues the numbers ˆ λ r ( m, n ) = m + n r .For m ∈ N • , the point ( m,
0) contributes one eigenfunction sin( mx ); for m, n ∈ N • ,the point ( m, n ) contributes two functions, sin( mx ) cos (cid:16) nyr (cid:17) and sin( mx ) sin (cid:16) nyr (cid:17) .Multiplicities of the eigenvalues also occur when the equation m + n r = m + n r has multiple solutions within the above range.As usual, we arrange the Dirichlet eigenvalues of C r in non-decreasing order, startingfrom the label 1, multiplicities accounted for,(6.2) 0 < λ ( C r ) < λ ( C r ) ≤ λ ( C r ) ≤ · · · . The purpose of this section is to prove Theorem 1.2, i.e. to determine the Courant-sharp eigenvalues in two specific cases r ∈ n , o whose eigenvalues have highermultiplicities. The corresponding cylinder are the orientation covers of the Möbiusband we studied in [5]. As in [11, 12], we could also consider other values of r , inparticular r close to zero or r very large. Remark 6.1.
As pointed out in the introduction, λ ( C r ) and λ ( C r ) are alwaysCourant-sharp, and any Courant-sharp eigenvalue λ k ( C r ) satisfies the inequality λ k ( C r ) > λ k − ( C r ) . Following Pleijel’s method, Section 2, we introduce Weyl’s counting function,(6.3) W C r ( λ ) = { j | λ j ( C r ) < λ } . According to Weyl’s law,(6.4) W C r ( λ ) = |C r | π λ + O ( √ λ ) = rπ λ + O ( √ λ ) , where |C r | denotes the area of the cylinder.6.2. Courant-sharp eigenvalues of C r , r ∈ n , o . According to [17, § 6], a do-main ω ⊂ C r , with area | ω | ≤ πr satisfies a Euclidean isoperimetric inequality,and hence its least Dirichlet eigenvalue satisfies(6.5) δ ( ω ) ≥ πj , | ω | . Lemma 6.2.
Let r ∈ n , o . Let λ k ( C r ) be a Courant-sharp eigenvalue of C r , with k ≥ π r . The following inequalities hold. (6.6) λ k ( C r ) k ≥ j , πr ≥ j , π ≈ . if r = , j , π ≈ . if r = 1 , (6.7) k − W C r ( λ k ( C r )) ≥ πr λ k ( C r ) − (2 r + 1) q λ k ( C r ) − . For r = 12 , π − j , ! λ k ( C ) − q λ k ( C ) − ≤ , (6.8) λ k ( C ) ≤ . , (6.9) For r = 1 , π − j , ! λ k ( C ) − q λ k ( C ) − ≤ , (6.10) λ k ( C ) ≤ . Proof.
The arguments are similar to those used in the proof of Lemma 4.1. Inequal-ity (6.6) follows from (6.5). The inequality in (6.7) follows from the description ofthe Dirichlet spectrum of C r . The other inequalities follow from (6.7) and (6.6). (cid:3) The following tables give the eigenvalues, the corresponding range of labels, and theratios λ kmin k min for C (left) and C (right). For a Courant-sharp eigenvalue, this ratioshould be greater than 1 . C , and greater than 0 . C .According to Remark 6.1, the eigenvalues λ and λ are always Courant-sharp.The table for C (left) shows that it only remains to analyze the eigenvalue 5 = λ ( C ) = λ ( C ). The associated eigenspace is generated by the eigenfunctionssin( x ) cos(2 y ) and sin( x ) sin(2 y ). Functions in this eigenspace have 2 nodal domains.This finishes the proof of Theorem 1.2 in the case r = .The table for C (right) shows that it only remains to analyze two eigenvalues 4and 5. The eigenvalue 4 = λ ( C ) is simple, with associated eigenfunction sin(2 x )which has two nodal domains. The eigenvalue 5 = λ ( C ) = · · · = λ ( C ) hasmultiplicity 4. The corresponding eigenspace is generated by the eigenfunctionssin(2 x ) cos( x ) , sin(2 x ) sin( x ) and sin( x ) cos(2 x ) , sin( x ) sin(2 y ) which, according to[5], turn out to span the second eigenspace E ( λ ( M )) of the square Möbius strip,whose orientation cover is C . Since the eigenfunctions in E ( λ ( M )) have two nodaldomains, it follows that the eigenfunctions in the eigenspace E ( λ ( C )) have at mostfour nodal domains. This finishes the proof of Theorem 1.2 in the case r = 1. Notethat one can give a precise analysis of nodal patterns in the eigenspace E ( λ ( C ))by using the same arguments as in [5, Section 4]. OURANT-SHARP EIGENVALUES 13 λ ( C / ) k min k max λ kmin ( C / ) k min λ ( C ) k min k max λ kmin ( C ) k min Table 6.1.
The first eigenvalues of C (left) and C (right) Appendix A. Classification of complete flat surfaces
Classifying complete, flat, Riemannian 2-manifolds is equivalent to classifying dis-crete groups of isometries acting freely on R . According to [18, p. 222-223], see also[26, Chap. 2.5], there are four types of such surfaces (up to a scaling factor). Eachone is given by its fundamental group, and the way it acts on the universal cover R in terms of Cartesian coordinates ( x, y ).(1) The cylinder (orientable surface),(A.1) ( x, y ) ( x + n, y ) , n ∈ Z ;(2) The torus (orientable surface),(A.2) ( x, y ) ( x + ma + n, y + mb ) , m, n ∈ Z , a, b ∈ R , b = 0 ; We point out that both the cylinder and the Möbius band mentioned here are infinite completesurfaces, not surfaces with boundary. (3) The Möbius band (non-orientable surface),(A.3) ( x, y ) ( x + n, ( − n y ) , n ∈ Z ;(4) The Klein bottle (non-orientable surface),(A.4) ( x, y ) ( x + n, ( − n y + mb ) , m, n ∈ Z , b ∈ R , b = 0 . For the flat tori and Klein bottles, we also refer to [8] (see 1.89, 2.25, 2.82, 2.83 and4.46), and [7].
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